International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2004 / Article

Open Access

Volume 2004 |Article ID 361614 | https://doi.org/10.1155/S0161171204210158

Werner Hürlimann, "Multivariate Fréchet copulas and conditional value-at-risk", International Journal of Mathematics and Mathematical Sciences, vol. 2004, Article ID 361614, 20 pages, 2004. https://doi.org/10.1155/S0161171204210158

Multivariate Fréchet copulas and conditional value-at-risk

Received18 Oct 2002

Abstract

Based on the method of copulas, we construct a parametric family of multivariate distributions using mixtures of independent conditional distributions. The new family of multivariate copulas is a convex combination of products of independent and comonotone subcopulas. It fulfills the four most desirable properties that a multivariate statistical model should satisfy. In particular, the bivariate margins belong to a simple but flexible one-parameter family of bivariate copulas, called linear Spearman copula, which is similar but not identical to the convex family of Fréchet. It is shown that the distribution and stop-loss transform of dependent sums from this multivariate family can be evaluated using explicit integral formulas, and that these dependent sums are bounded in convex order between the corresponding independent and comonotone sums. The model is applied to the evaluation of the economic risk capital for a portfolio of risks using conditional value-at-risk measures. A multivariate conditional value-at-risk vector measure is considered. Its components coincide for the constructed multivariate copula with the conditional value-at-risk measures of the risk components of the portfolio. This yields a “fair” risk allocation in the sense that each risk component becomes allocated to its coherent conditional value-at-risk.

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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